Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T19:05:56.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Distribution of the final extent of a rumour process

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Claude Lefevre  and
Philippe Picard
Claude Lefevre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université de Lyon I
*
∗ Postal address: Institut de Statistique C.P. 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
∗∗ Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

A rumour model due to Maki and Thompson (1973) is slightly modified to incorporate a continuous-time random contact process and varying individual behaviours in front of the rumour. Two important measures of the final extent of the rumour are provided by the ultimate number of people who have heard the rumour, and the total personal time units during which the rumour is spread. Our purpose in this note is to derive the exact joint distribution of these two statistics. That will be done by constructing a family of martingales for the rumour process and then using a particular family of Gontcharoff polynomials.

Keywords

MARTINGALESGONTCHAROFF POLYNOMIALS

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Short Communications
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 244 - 249
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by the Institut National de la Santé et de la Recherche Médicale under contract n° 921011.

References

Daley, D. J. and Kendall, D. G. (1965) Stochastic rumours. J. Inst. Math. Appl. 1, 42–55.CrossRefGoogle Scholar
Frauenthal, J. C. (1980) Mathematical Modelling in Epidemiology. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Lefevre, Cl. and Picard, Ph. (1990) A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 25–48.Google Scholar
Maki, D. P. and Thompson, M. (1973) Mathematical Models and Applications. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Picard, Ph. and Lefevre, Cl. (1990) A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 269–294.Google Scholar
Pittel, B. (1990) On a Daley-Kendall model of random rumours. J. Appl. Prob. 27, 14–27.Google Scholar
Sudbury, A. (1985) The proportion of the population never hearing a rumour. J. Appl. Prob. 22, 443–446.Google Scholar
Watson, R. (1988) On the size of a rumour. Stoch. Proc. Appl. 27, 141–149.Google Scholar